Bilinear embedding for real elliptic differential operators in divergence form with potentials

TL;DR

This paper introduces a straightforward Bellman function approach to establish a dimension-free bilinear estimate for elliptic divergence form operators with real coefficients and nonnegative potentials, valid across all p in (1,∞).

Contribution

It provides a novel, simple proof technique for bilinear estimates applicable to a broad class of elliptic operators with real coefficients.

Findings

Dimension-free constants in bilinear estimates.

Applicable for all p in (1,∞).

Works with nonsymmetric matrices.

Abstract

We present a simple Bellman function proof of a bilinear estimate for elliptic operators in divergence form with real coefficients and with nonnegative potentials. The constants are dimension-free. The $p$-range of applicability of this estimate is $(1,\infty)$ for any real accretive (nonsymmetric) matrix $A$ of coefficients.